Dynamics of coral-macroalgae interactions under crowding

We studied a planar ordinary differential equation (ODE) model for the benthic competition between coral, macroalgae, and algal turf on a reef, extending the classical model of Mumby, Hastings, and Edwards by a nonlinear, density-dependent coral mortality that accounts for crowding. The strength of crowding is set by an exponent $ \delta > 0 $ that reshapes the coral nullcline and enriches the bifurcation structure of the system. We established positive invariance of the biologically relevant region and the absence of periodic orbits, classified the three boundary equilibria together with their local stability, and reduced the coexistence problem to a single scalar equation whose shape (in particular, its concavity) controls the number and local stability of the interior equilibria. When crowding makes this equation non-concave, the reef can support three coexistence equilibria and become locally tristable. The grazing intensity $ g $ organizes the dynamics through two thresholds $ g_0 < g_1 $ determining the stability of the coral- and macroalgae-dominated states, and a further threshold $ g^\star $ at which two interior equilibria collide. We proved that the system undergoes a transcritical bifurcation at the coral-dominated state and, for $ \delta\in \mathbb{N} $, at the macroalgae-dominated state, together with a saddle-node bifurcation of interior equilibria, and we discussed the implications for coral reef resilience and hysteresis. We complemented these results with numerical simulations that illustrate the bifurcation sequence across the grazing regimes.